Published: September 27, 2011

r 2011 American Chemical Society 8261 dx.doi.org/10.1021/ma2010266 |Macromolecules 2011, 44, 8261–8269

ARTICLE

pubs.acs.org/Macromolecules

Microphase Separation and Phase Diagram of Concentrated DiblockCopolyelectrolyte Solutions Studied by Self-Consistent Field TheoryCalculations in Two-Dimensional SpaceYi-Xin Liu,*,† Hong-Dong Zhang,*,† Chao-Hui Tong,‡ and Yu-Liang Yang†

†Key Laboratory of Molecular Engineering of Polymers of Ministry of Education, Department of Macromolecular Science,Fudan University, Shanghai 200433, China‡Department of Physics, Ningbo University, Ningbo 315211, China

’ INTRODUCTION

Polyelectrolytes, among the most important classes of poly-mers, have been widely used in industry and they are attractingincreasing attentions in recent years due to their biorelatedapplications.1�3 When dissolved in polar solvents such as water,the ionization of chemical groups from polyelectrolyte chainbackbones results in charged polymers and small counterions.The long-range Coulomb interactions between these chargedspecies together with the short-range excluded volume interac-tions pose great challenges in theoretical study of polyelectrolytesystems. The challenges become more serious in particle-basedmethods such as the molecular simulation techniques, wherecomputationally expensive algorithms are unavoidable as long asthe long-range interactions are present.4 On the contrary, there isan efficient way to deal with the long-rangeCoulomb interactionsin field-based theories, where the electrostatic interaction isconverted to the short-range interaction by introducing the elec-trostatic potential into the partition function during the Hub-bard�Stratonovich transformation.5

The self-consistent field theory (SCFT), the most accuratetheory at the mean-field approximation level, is one of such field-based theories. It has become a standard technique for studyingmicrophase separations of neutral block copolymers owing to theefforts of Matsen and Schick,6 Drolet and Fredrickson,7 andRasmussen and Kalosakas8 on developing various highly efficientalgorithms or delicate screening techniques either in real space or

in spectral space.5,9 However, SCFT has not been widely used tostudy microphase separations in polyelectrolyte systems. Veryfew examples are available in the literature. The first systematicconstruction of SCF formalism for polyelectrolytes is given byBorukhov et al. in the middle 1990s, where a set of SCFTequations and mean-field free energies were derived for poly-electrolytes with various charge distributions in good solventsusing a path integral formulation.10,11 Random phase approxima-tion (RPA) has been performed to calculate the monomer�monomer structure factor S(q). Shi and Noolandi generalizedthe theoretical framework developed by Borukhov et al. to multi-component polyelectrolyte systems.12 This approachwas success-fully applied to study the interface of a simple single-componentpolyelectrolyte solution. Wang and co-workers further extendedthe SCFT of polyelectrolytes to block copolyelectrolytes andposition-dependent dielectric constant.13 The lamellar phaseof a symmetric diblock polyelectrolyte solution has been exam-ined in detail by solving the SCFT equations numerically. Kumarand Muthukumar applied the SCFT calculations to investigatethe dependence of the counterion distribution on the interac-tion parameter for the lamellar phase of a diblock copoly-electrolyte.14 The transition boundaries of the disorder-lamellar

Received: May 5, 2011Revised: September 7, 2011

ABSTRACT: The self-consistent field theory (SCFT) is applied to the microphaseseparation of concentrated solutions of weakly charged polyelectrolytes. The general-ized Poisson�Boltzmann equation describing the electrostatic interactions at themean-field level is numerically solved by a full multigrid algorithm, which enables oneto solve the SCFT equations of polyelectrolyte systems in real space as efficient asneutral polymer systems. To demonstrate the power of the real-space numericalscheme, we consider a diblock copolyelectrolyte consisting of a charged block and aneutral block in two-dimensional space. The phase diagram in the Flory�Hugginsinteraction parameter—the composition space is constructed by numerical calcula-tions. The density distribution of polymer segments, the counterions, and the netcharge of the ordered structures, namely the lamellar phase and the hexagonal phase,are intensively examined. The effects of the interaction parameter and the degree ofionization are examined carefully. The numerical scheme can be easily extended to 3Dcalculations, various chain architectures, various charge distribution models, and other statistics chain models such as the worm-likechain model without losing any computational efficiency.

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transition, the cylinder-lamellar transition, and the sphere-cylindertransition were calculated based on RPA analysis. Recently, Yanget al. extended the reciprocal-space SCFT method originallydevised by Matsen and Schick6 to the polyelectrolyte systems,where phase diagrams of A�B diblock copolyelectrolytes andA�B�A triblock copolyelectrolytes, both with A blocks beingcharged, were calculated.15 To date, however, there is still lack ofan efficient real-space SCFT method for studying the phaseseparation of charged polymers other than in one-dimensional(1D) space. This is mainly due to lack of a time-efficient algorithmto solve SCFT equations when the generalized Poisson�Boltzmann(PB) equation is involved. Nevertheless, real-space SCFT methodsdo not require any a priori knowledge of symmetry, whichbecomes a great advantage when one wants to search new phasestructures. In addition, the morphology of the equilibrium phaseis a natural consequence of the real-space calculation, whichprovides important information for both theoretical and experi-mental researchers. For instance, the distribution of solvent mole-cules and counterions can be directly obtained by analyzing themorphology.

In this article, by introducing the full multigrid algorithm(FMG) to solve the PB equation, we develop a real-space numericalscheme which can solve the SCFT equations of the charged poly-mer system as efficient as the neutral polymer system. The num-erical scheme is highly extensible and it is feasible to solve theSCFT equations in both two-dimensional (2D) and three-dimensional (3D) spaces. In principle, polyelectrolytes with anyarchitecture and with any number of blocks, each of which can beeither charged or neutral, can be treated using this numericalscheme. In particular, a concentrated solution of a diblockcopolyelectrolyte consisting of a negatively charged (A) blockand a neutral (B) block has been considered in this work. A phasediagram was constructed based on 2D SCFT calculations for thechosen system. In addition to the lamellar phase (LAM), thehexagonally packed cylinder phase (HEX) has been analyzed indetail.

’THEORETICAL METHODS

A. SCFT Formalism. Here, we sketch a general theoreticalframework for a concentrated solution of diblock copolyelectro-lytes containing nC polymer chains and nS solvent molecules withor without salts. Each A�B copolymer has N total statisticalsegments with the volume fractions f and 1� f for A and B blocks,respectively. It is supposed that the polymer segment and thesolvent molecule have the same density F0, and the volume ofsmall ions is ignored. We use subscripts A, B, S, +, and � invariables to denote A segments, B segments, solvent molecules,cations, and anions, respectively. The valences of charged speciesare represented by integer variables zi (i = A, B, +, and �).To construct the statistical field theory for this charged system,

we adopt the continuous Gaussian chain model. The smearedchargemodel is introduced to describe the charge distribution. Inthis model, charges are assumed to distribute uniformly along thechain contour. One example that should be well described by thismodel is the strongly dissociating polyelectrolyte, such as poly-(acrylic acid) (PAA). In addition, the primitive model is used todescribe the electrostatic interactions between two point chargesmediated by the solvent. The solvent is considered as a con-tinuous medium with a dielectric constant. With the aboveconsiderations, the free energy per chain of the system at volume

V and temperature T (in units of kBT where kB is the Boltzmannconstant) is given by13

F ¼ 1V

Zdr ½N∑

K∑L 6¼K

χKLϕKϕL � ∑KwKϕK

� ηð1� ∑KϕKÞ� �

1V

Zdr

ε

2j∇ψðrÞj2

� ϕ̅C lnQCϕ̅C

�N∑Mϕ̅M ln

QMϕ̅M

ð1Þ

Here, ϕK (ϕL) are density fields normalized by F0, and ωK arecorresponding conjugate potential fields introduced to exert inter-actions on species K (L) (K, L = A, B, and S);ψ is the electrostaticpotential field; η is a Lagrange multiplier that ensures the incom-pressibility of the system; χKL denote the Flory�Huggins interac-tion parameters between species K and L; ε represents the dielectricconstantwhich is invariant across thewhole system, and it is rescaledby 8π2ε0F0e

2b2/3 with ε0 the dielectric constant of vacuum, e theunit charge, and b the length of a statistical segment (Kuhn length);ϕ̅C � nCN/F0V and ϕ̅M � nM/F0V are the volume-averageddensities for the block copolymer and for species M (M = S, +,and �), respectively. Note that the spatial quantities in eq 1 arerescaled by the radius gyration of an unperturbed Gaussian chainRg = b(N/6)

1/2, i.e. r/Rg f r and V/Rgd f V with d the

dimensionality of the system. In eq 1, QC is the normalizedsingle-chain partition function for copolymer and QM are thenormalized single-particle partition functions for species M.Minimization of the free energy (eq 1) with respect to ϕK

leads to the equations of equilibrium potential fields wK(r) =N∑L6¼KχKLϕL(r) + η(r). Similarly, the equilibrium density fieldscan be obtained by minimizing the free energy with respect to wj(j = A, B, S, +, and �). They are given by ϕA(r) = (ϕ̅C/QC)

R0f ds

qC(r,s)q*C(r,1 � s), ϕB(r) = (ϕ̅C/QC)Rf1ds qC(r,s)q*C(r,1 � s),

ϕS = (ϕ̅S/QS) exp[�wS(r)/N], and ϕ( = (ϕ̅(/Q() exp[�z(ψ-(r)]. q(r,s) is a forward chain propagator which corresponds tothe probability of finding the end segment of the polymer chainof length sN starting from the A block at location r, which satisfiesthe modified diffusion equation

∂qðr, sÞ∂s

¼ ∇2qðr, sÞ � ½wAðrÞ þ zAαANψðrÞ�qðr, sÞ, if s e fN

∇2qðr, sÞ � ½wBðrÞ þ zBαBNψðrÞ�qðr, sÞ, if s g fN

(

ð2Þwith the initial condition q(r,0) = 1. αA andαB in eq 2 denote thedegrees of ionization of A and B blocks, respectively. The degreeof ionization is defined as the number of unit charges perstatistical segment. The backward chain propagator q*(r,s)initiated from the end of the B block satisfies a diffusion equationsimilar to eq 2.13 The normalized single-chain partition functionis given by QC = (1/V)

Rdr q(r,1), while the normalized single-

particle partition functions are given by QS = (1/V)Rdr exp-

[�wS(r)/N] and Q( = (1/V)Rdr exp[�z(ψ(r)].

To complete the set of SCFT equations, one needs tominimize the free energy with respect to ψ to find the electro-static potential in equilibrium. It is straightforward to show thatthe equilibrium electrostatic potential satisfies the followinggeneral Poisson�Boltzmann equation:

∇2ψðrÞ ¼ �Nε½αAzAϕAðrÞ þ αBzBϕBðrÞ þ zþϕþðrÞ þ z�ϕ�ðrÞ�

ð3Þ

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B. Numerical Method. The above mean-field equations canbe numerically solved by a quasi-Newton method with fastconvergence and remarkable accuracy.13 However, the quasi-Newton method involves an inversion of a Jacobian matrix,which is an algorithm with O(M3) computational complexitywhere M is the scale of the problem. The scale of a problem isdefined as the total number of grids that the space has beendiscretized into. For a 3D problem in a cubic cell,M is equal to L3,where L is the number of discrete points in each side of the cell.Therefore, a huge amount of operations, L6 for 2D problems andL9 for 3D problems, are required, so that only one-dimensionalproblems are feasible in practice based on the quasi-Newtonmethod. To overcome this difficulty, here we turn back to thecontinuous steepest descent method5 which has been proven tobe a simple but effective strategy. The set of SCFT equations aresolved in a similar way as that described by Drolet and Fredrickson7

except the PB equation (eq 3). First, fields wA, wB, wS, and ψ areinitiated by random numbers or by preset values. Then themodified diffusion equations for both forward and backwardchain propagators are solved by utilizing a pseudospectral algo-rithm with nearly ideal computational complexityO(NsM lnM).

8

After that, the density fields ϕj are evaluated form the knownpotential fields. These densities are used to produce new fieldswA, wB, and wS. The auxiliary field η is updated according toηnew = ηold + λη(1 � ϕA � ϕB � ϕS) where λη is a relaxationparameter controlling the strength of the incompressibility.Given that we have found an efficient way to solve the PB equation,the new electrostatic potential field can be constructed by a linearmixing of the solution of the PB equation and the old field. Aboveprocedures form a typical computational unit of the continuoussteepest descent strategy. The computational unit is then exe-cuted repeatedly until some stop criteria are eventually met.If the solution of the PB equation is not taken into account, the

most time-consuming step during each iteration is the solution ofdiffusion equations which involves at least O(NsM ln M) opera-tions whereNs = 1/Δs is number of points the chain contour hasbeen discretized with Δs the contour step. The PB equationreduces to a Poisson equation if the densities in the right-handside of eq 3 are replaced by those of the previous iteration.Poisson equation is perhaps the most well-known linear differ-ential equation. A large number of algorithms are available tosolve it numerically, the complexity of which ranges from O(M)toO(M3). In this work, we introduce the full multigrid algorithm(FMG) which is the most efficient one with optimal O(M)complexity.16�18 In principle, only O(M) operations are neededto solve eq 3 with FMG, which is negligible compared to that ofsolution of diffusion equations as long as Ns ln M . 1, which isfulfilled in most cases. Remarkably, charged polymer systems cannow be numerically solved in a time comparable to that of neutralpolymer systems. In other words, the FMG enables us tocalculate the phase behavior of charged polymer systems in bothtwo-dimensional (2D) and three-dimensional (3D) space.One of drawbacks of the multigrid method is that there is no

such standard multigrid solver available. It is still a nontrivial taskto implement multigrid method for a given problem. Only afterthe property of the differential equation has been well exploited,the proper smoothing operator, restriction operator, and inter-polation operator, the most important components for a multi-grid program, can be chosen. In this work, we have successfullyimplemented the FMG for eq 3. Each component in the programhas been carefully tuned to give ideal performance. Specifically,we choose a red-black Gauss�Seidel relaxation scheme as the

smoothing operator, bilinear interpolation as the interpolationoperator, and full-weighting restriction as the restriction operator.17

The periodic boundary condition is imposed on all levels. Thecorrectness and accuracy of the implementation are verified bysolving analytical-solvable Poisson equations by comparing numer-ical solutions with analytical solutions. To verify the efficiency ofthe FMG program, we perform a speed test on our FMGprogram and the software package MUDPACK-519 on the sameplatform. We find that our implementation is 10�15% fasterthan MUDPACK-5.C. Case Study: Charged-Neutral Diblock Copolymers in a

Salt-Free Solution. To demonstrate the power of the numericalmethod, we consider a salt-free solution of charged-neutraldiblock copolymers. In particular, we set N = 400, ϕ̅C = 0.8,zA =�1, z+ = 1, and χAS = χBS = 0 (the solvent is a good solventfor both A block and B block). Other parameters zB, αB, z�, andϕ̅� are all 0 for the B block is neutral and no salt is added to thesolution. By invoking the constraints of incompressibility andelectroneutrality, other volume-averaged densities given byϕ̅S = 0.2 and ϕ̅+ = 0.8αA f. With the above setup, we mainlyinvestigate the effects of following three parameters: f, χAB, and αA.Numerical SCFT calculations are carried out in a 2D cell with

periodic boundary conditions. The cell with physical sizes lx� lyis discretized into L� L lattices where L = 2m with a typicalm of 7.The lattice spacings along the x and y directions are determinedby Δx=lx/L and Δy=ly/L. The typical lattice spacing in ourcalculations is 0.03Rg. Note that here we intentionally use a muchsmaller lattice spacing than 0.1�0.2Rg which is typical for neutralpolymer systems. The lattice spacing is small because lx and lyshould be small to ensure that the cell contains only one or twoperiods of phase structure and L needs to be large because ofFMG. The smaller lattice spacing also improves the accuracy ofthe solution though it consumes more computational time. Thevalue of Ns used to discretize the chain length is fixed at 200. Aslong as the right-hand side of eq 3 is smooth enough (none ofN/ε,αA, and χAB is too large), the continuous steepest descent schemeis stable if the relaxation parameters are carefully chosen. Aworking group of relaxation parameter are 0.05 for updating thepotential fields wA, wB and wS, 0.1 for updating the electrostaticpotential fieldψ, and 10 for updating the auxiliary field η. Some-times smaller relaxation parameters are needed to stabilize thealgorithm, e.g., when the system is near phase boundaries, or theinteractions (either Flory-type or electrostatic) are strong. Inpractice, the difference of mean-field free energies between twoconsecutive iterations and the total residual error13 are bothmonitored. We choose the total residual error being smaller than10�9 as a stopping criterion. Typically, the stop criterion is metafter about 5,000 iterations. For calculations close to phaseboundaries, 50 000 or more iterations are needed to reach thecomparable precision. The typical time needed to complete onesuch calculation is estimated to be 5000� 0.35/60 s≈ 30min ona single 2.5 GHz CPU core.

’RESULTS AND DISCUSSION

A. Asymmetric Phase Diagram. As described previously, thehighly efficient FMG enables us to treat charged polymer problemsas neutral polymer problems at least in the sense of the numericalcomputation. It becomes a routine work to construct phase dia-grams of charged polymer systems in the χABN ∼ f parameterspace by SCFT calculations. The phase diagram of a concentratedcharged-neutral diblock copolyelectrolyte solution is presented in

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Figure 1a, where we have successfully determined phase bound-aries for three phases, namely the disordered phase, the cylind-rical phase (HEX), and the lamellar phase (LAM), while thoseintrinsically 3D mesophases such as bcc spheres, close-packedspheres, and bicontinuous gyroid is unreachable in 2D calculations.To locate phase boundaries in relatively high accuracies (0.001

for composition f and 0.1 for χABN), we have devised a two-stagescan scheme. At the first stage, the combinatorial screeningtechnique7 with slight modifications is used to roughly determinethe phase boundaries. The set of SCFT equations are solved inreal space with random initial conditions. The calculations willconverge to either stable or metastable phases. For high χABNvalues, we fix χABN and perform a scan from low f to high f with astep of 0.1 in a square cell. This scan path is parallel to the abscissaaxis (f) and thus is called a parallel scan. Along the scan path, it isexpected that we will encounter a disorder-to-cylinder transition(ODT), followed by a cylinder-to-lamellar transition (OOT),and then followed by two similar transitions in the reverse order.For lower χABN values, χABN instead of f is varied from low valueto high value with a step of 0.5. This kind of scan is called avertical scan since the scan path is parallel to the vertical axis(χABN). During the vertical scan, it will traverse an ODT fol-lowed by an OOT (sometimes the OOTwill be bypassed as longas the gap between LAM and HEX is smaller than the scan stepsize). From these scans, we can roughly determine the phaseboundaries.At the second stage, the scans are narrowed to the region close

to phase boundaries determined at the first stage. The SCFT

calculations are performed under given initial conditions. Thefields wA(r) and wB(r) are initialized by random numbers if thedisorder phase is expected, while they are initialized by a one-period 2D lamellar pattern if LAM is expected, and they areinitialized by a two-period 2D hexagonal lattice pattern if HEX isexpected. It should be noted that HEX is not compatible with thesquare cell due to the symmetry. Therefore, we will use arectangle cell with lx: ly = 2:(3)

1/2 to perform calculations whenHEX is expected. For all calculations, the field wS(r) is initializedby random numbers and the field ψ(r) is initialized by 0. Todetermine the equilibrium phase for each point in the phasediagram, we also systematically vary cell sizes to eliminate the sizeeffect of the simulation cell. The equilibrium phase at each point(f, χABN) is then assigned to the calculated phase structure withthe lowest free energy. Each symbol in Figure 1a corresponds tosuch a determination of the equilibrium phase. The exact transitionphase boundary should lie between two neighboring symbols ofdifferent phases, whose gap clearly determines the accuracy of thephase boundary.To get a rough idea about how good our SCFT calculations

are, we plot the spinodal line (the dashed line) of the concen-trated solution predicted by random phase approximation (RPA)in Figure 1a. It can be seen that the binodal line (the phaseboundary between the disorder phase and HEX) determined byour SCFT calculations and the spinodal line predicted by RPAtheory are so close that they are indistinguishable. This behavioris typical for diblock copolymer melts,20 and it also occurs incharged diblock copolymer melts.14 In the latter case, the binodalline and the spinodal line get closer and closer when either thedegree of polymerization or the degree of ionization increases.Since our system is rather similar to the reported one and thesimilar behavior was actually observed, it implies that our SCFTcalculations are indeed a valid tool for predicating of phase diagrams.It is worth noting that the spinodal line in this article is

obtained by conducting a stability limit analysis on the partialstructure factor, SAA(q) = ÆδϕA(q)δϕA(�q)æ in Fourier space,which characterizes the spatial correlation of the concentrationfluctuations of A blocks. The partial structure factor SAA(q) hasthe form21

NSAA�1ðqÞ ¼ NSneutral�1ðqÞ þ ðαANÞ

2

εx þ ðαANÞf ϕ̅Cð4Þ

where x = q2Rg2. The first term in the right-hand side of the above

equation contains the contributions of all interactions except theelectrostatic interaction, which is identical to the correspondingpartial structure factor of the neutral diblock copolymer solution.This term can be derived from the linear response theory

Sneutral�1ðqÞ ¼

1=ϕ̅C þ ϑNht þ ðχABNÞh12�ðχABNÞ½ðχABNÞ þ 2ϑN�ϕ̅Cðh1h2�h122=4ÞN½h1 þ ϑNϕ̅Cðh1h2 � h122=4Þ�

ð5Þwhere h1 is the Debye function defined as h1 = h(f,x) =2(fx + e�fx � 1)/x2, and h2 = h(1 � f,x), ht = h(1,x), andh12 = ht � h1 � h2. In eq 5, ϑ = 1/ϕ̅S � 2χPS is an excludedvolume parameter related to the short-range interactionsbetween solvent molecules and polymeric segments. The secondterm in the right-hand side of eq 4 contains the contributionoriginated from the long-range Coulomb interactions at theDebye�H€uckel level.

Figure 1. Phase diagrams of a concentrated salt-free solution of diblockcopolyelectrolytes obtained by 2D SCFT calculations. The degree ofionization αAN is fixed at 20. Key: red squares, LAM; green filled circle,HEX; black empty circle, disordered phase; dashed line, the spinodal linepredicted by RPA theory; solid line, the HEX�LAM phase boundariesfrom 2D SCFT calculations.

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The partial structure factor SAA(q) possesses a maximum at acertain wavelength qs whose value is independent of the segrega-tion effects. However, the shape of SAA(q) strongly depends onthe segregation effects. For neutral polymers, the controllingparameter is the product χABN rather than the interaction para-meter χAB alone, while for charged polymers, there is an ad-ditional parameter the product of the degree of ionization and thedegree of polymerization αAN due to the electrostatic interac-tion. Equations 4 and 5 clearly reveal these two kinds of dependen-cies. There is a critical value of χABN, (χABN)s, beyond which thepeak in SAA(q) diverges at q = qs. (χABN)s is thus identified as thestability limit of the system. Unlike neutral polymers, thespinodal line in the (χABN)s ∼ f plot is no longer universal butdepends on the value of αAN. In other words, the spinodal lineshould remain unchanged as long as αAN is fixed no matter howαA and N are varied independently. Therefore, it can be con-sidered as another type of universality specific to charged polymersystems. This assertion should be safely extended to all bound-aries. We would like to emphasize that this kind of universality isan important feature of charged polymer systems that seems tobe overlooked for years. Besides of χABN and αAN, the dielectricconstant ε is another interesting parameter that can influence thephase diagram of charged polymer systems as can be seen fromthe second term in the right-hand side of eq 4.As the validity of the 2D SCFT calculation is established, it is

ready for us to examine some properties of the phase diagram.The most obvious difference on the phase diagram between thecharged polymer and the neutral polymer is that the orderedphase region of the former phase diagram is pushed upward.Taking f = 0.5 as an example, the interaction parameter of thecritical point is 24.65 ( 0.05 as compared to 10.495 for neutraldiblock copolymer melt and 13.119 for neutral diblock copoly-mer solution. The increase of the critical interaction parametermeans that the miscibility of A and B blocks are enhanced byintroduction of charges. Marko and Rabin suggested that theenhancement of miscibility is mainly due to the release of counter-ions of the charged polymer chains which significantly increasesthe mixing entropy.22 It should be pointed out that the electro-static interaction among charged species have less important effecton the enhancement of miscibility than the release of counterions.Another difference between the charged polymer system and

the neutral polymer system is that the phase diagram of thecharged polymer system is asymmetric. In Figure 1a, the asymmetryis not that obvious. RPA theory predicts that the degree ofasymmetry will enhance with the increase of αAN.

14 To illustratethe asymmetry clearly, the region near the critical point is am-plified in Figure 1b. It can be seen that the composition f at thecritical point shifts to 0.53 in this particular case (RPA theorypredicts 0.536) as compared to 0.5 for neutral diblock copoly-mers. Moreover, the spinodal line, the binodal line, and the OOTboundary (solid lines in Figure 1) are all asymmetric. Theseasymmetries lead to the shift of HEX and LAM regions of thephase diagram toward high f end. In particular, theHEX region tothe left of the critical point is enlarged and that to the right iscontracted as seen in Figure 1a. This behavior is similar to thecase in which the statistic segment length of the A block is largerthan that of the B block.23,24 In contrast to the enhancement ofmiscibility, the asymmetry of the phase diagram is mainly causedby the electrostatic interaction.These results may be very helpful for experimental studies. For

diblock copolymers with one charged block, the critical point nolonger locates at f = 0.5. Consequently, the equilibrium phase

near f = 0.5 is not necessary LAM. This fact may explain whyHEX and gyroid were often observed near f = 0.5 in poly(styrene-sulfonate-b-methybutylene) (PSS�PMB) copolymers wherePSS is the charged block and PMB is the neutral block.25,26

The PSS blocks were prepared by randomly sulfonating PSblocks with a sulfonation level ranging from 10% to 50%. Sincethe sulfonic acid group is strongly acidic, PSS should be welldescribed by the smeared charge model. Therefore, one can takethe sulfonation level as the degree of ionization of PSS blocks.The degree of ionization in these samples is much higher than 5%used in our SCFT calculations, which should lead to much largerdegree of asymmetry of the phase diagram than that shown inFigure 1a. Hence HEX will be the most possible equilibriumphase in the region between f = 0.45 and f = 0.5. In 3D space as inthe experiments, gyroid phase is also possible. In light of thediscussion on the universality of the controlling parameters(χABN and αAN), it is important to note that the phase diagramsreported in ref 26 are very complicated since the product αANwas not fixed in a same phase diagram. These phase diagrams canconsist of phase boundaries with very strange shapes, dependingon how N and αA are varied.On the basis of the above discussion, it can be concluded that

our SCFT calculations qualitatively agree with the experimentalfindings. However, one should be aware that the Gaussian chainmodel may be no longer adequate for describing the chain statisticswhen the charge density on the chain becomes large (i.e., αA islarge). The large charge density on the chain will inevitably rigidifythe chain, where the worm-like chain model should be more ap-propriate. Moreover, when the charge density on the polyelec-trolyte chain exceeds some critical value, Manning condensationwill occur and the charge density on the chain should be renorma-lized. To note further, our approach is in essential at the mean-field level, which means that the approach breaks down whenmultivalent ions are introduced because the correlation betweenthose counterions can no longer be ignored. In this work, how-ever, we are only interested in the cases where the Gaussian chainmodel still works. Meanwhile, only univalent counterions areconsidered. Incorporating the worm-like chain model into SCFTfor charged polymer system should be the future work.B. Structures of Ordered Phases. Although RPA theory can

predict the equilibrium phase, it is unable to provide informationabout the underlying structures of the equilibrium phases, e.g.,the density distributions of the A segments, the B segments, thesolvent molecules, and the charged components (the free counter-ions and the charges distributed on the A blocks). On the contrary,the structure information is a direct consequence of the SCFTcalculations. In literature, real-space 1D SCFT calculations areperformed, where only 1D structure, i.e. lamellar phase, can beobserved. In this work, real-space 2D SCFT calculations allow usto study an additional structure: HEX.Figure 2 shows typical density distributions of the polymer

segments and the solvent molecules, accompanied by the corre-sponding density distributions of the neutral diblock copolymersolution for the sake of comparison. It can be seen that thelamellar phase is resulted for the symmetrical charged diblockcopolymer solution as expected. The system undergoes a micro-phase separation into two distinct domains: the neutral-block-rich domain (the left regions of Figure 2, parts a and c) and thecharged-block-rich domain (the right regions of Figure 2, parts aand c). Similar to the neutral diblock copolymers, the densitydistributions in the A-rich domain and in the B-rich domain areperfectly symmetric. However, the interfaces between the A-rich

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domain and the B-rich domain in the neutral diblock copolymersolution are much sharper than those in the charged diblockcopolymer solution under the same condition, indicating that themiscibility of A and B blocks is enhanced by introducing charges.More interestingly, the solvent molecules not only prefer gather-ing in the interfacial region as they do in the neutral diblockcopolymer solution but also prefer staying in the charged-block-rich domain. The solvent density is peaked at the center of theinterface (f = 0.5) and decays more rapidly into the charged-block-rich domain than the neutral-block-rich domain, unlike theneutral diblock copolymer solution where the density decayssymmetrically into both sides. Consequently, the solvent densityis higher than the averaging value ϕ̅S in the interfacial region andin the charged-block-rich domain. For the charged polymer system,we also observed that the peak of the solvent density at themiddle ofthe interface is pronounced. Meanwhile, the decay lengths areshortened. These observations suggest stronger segregation of thesolvent molecules between the charged-block-rich domain andthe neutral-block-rich domain. This kind of unbalanced distribu-tion of solvent molecules in one lamellar period is first demon-strated here. The fluctuation of the solvent molecules may havesome important consequences in practical situations such as thetransportation of small ions.Next, let us have a look at the charge density distribution of the

free counterions and the net charge density distribution, whichare shown in Figure 3a and 3b, respectively. The charge densitydistribution of the free counterions is equivalent to the densitydistribution of the free counterions since z+ = 1. The net chargedensity is a local variable measured by ϕe(r)=�αAϕA(r)+ϕ+(r). Itcan be seen that the counterions can dissociate from polymer chainsand diffuse into the neutral-block-rich domain. As a consequence,

the overall charge in charged-block-rich domain is negative and itis positive in the neutral-block-rich domain. At the interfaces, thenet charge density tends to be zero. In order to view the detailedvariation of charge densities, we plotted the charge density profilesalong the layer normal in Figure 2c. As shown in Figure 2c, thecounterion charge density in the charged-block-rich domain isabove the averaging value ϕ̅+ = 0.02, while it is below this value inthe neutral-block-rich domain. In contrast to the density dis-tribution of the polymer segments, the charge density profiles areasymmetric. The charge density profile of the free counterionsexhibits a broader peak in the neutral-block-rich domain than thepeak in the charged-block-rich domain, whereas the net chargedensity profile behaves in an opposite manner. Although theelectrical double-layers are not obvious in the net charge densityprofile, it will become more obvious when either χABN or αANincreases. In the net charge density profile, one can still identify afaint peak located at x/lx≈0.65 rather than at x/lx = 0.5 where thepeak of the solvent density locates, implying that the electricaldouble-layer is actually formed. The decay of the peak (the oneat x/lx≈ 0.65) into the charged-block-rich domain is so slow thatit interferes with the one decaying from the peak of the adjacentinterface (the one at x/lx ≈ 0.85), leading to a broad peak atx/lx = 0.75 (see the inset of Figure 3c). On the other hand, in theneutral-block-rich domain, the peak of the electrical double-layerin this domain is so broad that it overlaps the peak of the adjacentinterface, leading to an overall broad peak. It is worth mentioningthat we have also studied the dependencies of the charge densitieson χABN and αAN. The results are essentially the same as those

Figure 3. Charge density distribution of the free counterions (a) andthe net charge density distribution (b) calculated by SCFTwith the sameparameters as those in Figure 2. The charge density of the free coun-terions is normalized by the averaging value ϕ̅+ and rescaled via ϕ̂+(r) =ϕ+(r)/max(ϕ+(r)). The net charge density distribution is plotted in redfor negative charges and in green for positive charges. It is normalized byϕ̂e(r) = |ϕe(r)|/ϕ̅+. (c) Charge density profile of the free counterions(black solid line) and the net charge density profile (blue dash line)along the layer normal. The inset is a zooming in plot of the net chargedensity profile.

Figure 2. Density distributions of type A (red) and B (blue) segments(left columns), and solvent molecules (right columns) for charged-neutral diblock copolymer solutions (top row) and the same solutions ofneutral diblock copolymers (bottom row) with f = 0.5, χABN = 35. Thedegree of ionization for charged blocks is αAN = 20. The cell sizes are(a, b) 2.7� 2.7 Rg2 and (c, d) 4.4� 4.4 Rg2. For all figures, the purer andthe brighter the color is, the higher the corresponding concentration is.This applies to all following figures. The concentration of solvent hasbeen rescaled according to ϕ̂S(r) = [ϕS(r)�min(ϕS(r))]/[max(ϕS(r))�min(ϕS(r))].

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reported in ref 13 and 14 despite the fact that their results wereobtained from 1D SCFT calculations.When the composition f increases to 0.7, the charged-neutral

diblock copolymer solution tends to phase separation into HEXphase with intermediate interaction parameters (30

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(bear positive charges) and the limited increase of the density ofthe A segments (bear negative charges) give a drop of the netcharge density in the domain. The drop of the net charge densityis clearly illustrated by the dark regions inside the green disks asshown in Figure 6d. The dark regions become darker and largerwith increasing χABN. It has been proposed that all free coun-terions will be confined to the charged domains in the segrega-tion limit (χABN f ∞).14 Therefore, the dark regions willeventually occupy the whole area of the charged-block-rich

domains in the segregation limit, which results in electroneu-trality everywhere.Analysis of the charge density profiles gives the same extra-

polation. Figure 7a shows the charge density profile of the freecounterions and the net charge density profile along the xdirection in Figure 6a�d, where one can easily identify the elec-trical double-layers at the interfaces. The peaks in the electricaldouble-layers become sharper as the miscibility of the A and Bsegments reduces. It is expected that in the segregation limit,these peaks becomes so sharp that their widths tend to 0, and thepeaks with positive intensity cancels out the peaks with negativeintensity. Consequently, the net charge density profile is stretched toa flat line parallel to the abscissa axis with ϕe(r) = 0.As we already know, contrary to the interaction parameter

χABN, the increase of the degree of ionization αAN enhances themiscibility of the charged blocks and the neutral blocks. One mayintuitively guess that αAN will have an exactly opposite effect onthe charge density distribution of the free counterions and the netcharges density distribution as χABN does. However, the casehere is a little complicated since the degree of ionization directlycontrols the averaging value of the charge density of the freecounterions via ϕ̅+ = 0.8f αA. After the completion of themicrophase separation, the charge density of the free counterionsshall spatially fluctuate around the averaging value. To comparethe charge density distribution of the free counterions at differentdegrees of ionization, we scale them by the averaging value ϕ̅+ (seethe caption of Figure 7b). Surprisingly, in the range of small degreesof ionization (αAN e 0.04), increasing αAN leads to strongersegregation of counterionswhile the segregation of A andBblocks isweakened. Meanwhile, the electrical double-layers in the net chargedensity profiles are enhanced. The underlying mechanism of thisunexpected phenomenon is not fully understood at present. Wepropose that the increase of αAN must have two effects: one is toenhance the miscibility of the charge blocks and the neutral blocks;the other is to enhance the separation of counterions between thecharged-block-rich-domain and the neutral-block-rich domain. Forsmall degrees of ionization, the enhancement of the separation ofcounterions overwhelms the enhancement of miscibility, leading tothe strong separation of the counterions. While for large degrees ofionization, the enhancement of the miscibility will take over theenhancement of the separation of counterions, so that the as-expected effect of αAN should be restored (see the profiles with20 < αAN < 28 in Figure 7b). It is found that when αAN > 28, theequilibrium phase turns out to be homogeneous.

Figure 6. Charge density distribution of the free counterions (odd columns) and the net charge density (even columns) depend on the interactionparameter χABN (top row) and the degree of ionization αAN (bottom row) with f = 0.7. χABN are (a, b, e-h) 35 and (c, d) 50. αAN are (a�d) 20, (e, f) 4,and (g, h) 24. The cell sizes are (a, b) 6.35 � 5.50 Rg2, (c, d) 7.27 � 6.30 Rg2, (e, f) 8.66 � 7.50 Rg2, (g, h) 5.89 � 5.10 Rg2. The charge density ofcounterions and the net charge density are rescaled as described in Figure 3.

Figure 7. The dependences of the charge density profile of the freecounterions (solid line) and the net charge density profile (dashed line)along the x direction on (a) the interaction parameter χABN and (b) thedegree of ionization αAN. αAN = 20 in part a; χABN = 35 in part b. Thecharge density of the free counterions is scaled according to ϕ̂+(r) =ϕ+(r)/ϕ̅+ in part b. All other parameters are the same as those inFigure 6.

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’CONCLUSIONS

In this study, we developed a highly efficient numerical schemefor solving the SCFT equations containing a general Poisson�Boltzmann equation in real space. The introduction of the fullmultigrid algorithm drastically reduces the computational timefor the solution of the PB equation. The time consumption isnegligible in comparison with the solution of the modified diffusionequations. The power of this technique was demonstrated by per-forming a series of real-space 2D SCFT calculations to construct thephase diagram of a concentrated charged-neutral diblock copolymersolution.

The phase diagram of the charged-neutral diblock copolymersolution is asymmetric. The homogeneous phase is stabilized byintroducing dissociable charges onto the polymer backbones.Increasing the degree of ionization (αAN) enlarges the parameterspace of the homogeneous phase. Meanwhile, the critical pointshifts to higher compositions (f). Consequently, the ODT phaseboundary and the OOT phase boundaries are also lose theirmirror symmetry. These changes of the phase diagram lead tolarger parameter space for the HEX phase near f = 0.5, which canbe used to explain the recent experimental results. Our numericalstudies are consistent with the RPA theory, the 1D SCFT cal-culations, and the reciprocal-space SCFT calculations. The detailedstructures of the ordered phases were examined. And the effectsof the interaction parameter (χABN) and αAN on the densitydistributions of the polymer segments, the solvent, and thecharge components were analyzed systematically. It was foundthat the increase of χABN and the decrease of αAN have similareffects on the density distributions of the polymer segments andthe solvent, while they influence the charge density distributionsin a quite different manner.

Owing to the fact that the implementation of FMG is indepen-dent from the implementation of the continuous steepest descentscheme, the numerical approach developed in this work is highlyextensible. For example, it is straightforward to apply it to exploreordered structures in 3D space, to study other chain architec-tures, other charge distribution models, and other statistics chainmodels such as the worm-like chain model (with this model,study of the strongly charged polyelectrolytes becomes possible).Furthermore, many other parameters, such as the dielectric con-stant, the concentration of the added salts, the concentration ofthe polymers, are all ready to be examined without altering theimplementation. In addition, it is also valuable to analyze thedomain spacings of the ordered structures, which can be directlyobtained in real-space SCFT calculations. The dependences ofthe domain spacings on various controlling parameters will bereported elsewhere. We hope that the real-space SCFT tech-nique devised in this work shall advance the SCFT study of theordered structures in charged polymer systems on surfaces (2D),in membranes (2D), and in bulk (3D).

’AUTHOR INFORMATION

Corresponding Author*E-mail: liuyxpp@gmail.com (Y.X.L.); zhanghongdong@gmail.com (H.D.Z.).

’ACKNOWLEDGMENT

The authors are grateful to Prof. An-Chang Shi andDr. Jian-FengLi for their discussions. This project is sponsored by ShanghaiPostdoctoral Scientific Program (2011) and supported by the

National Nature Science Foundation of China (Grant Nos20874019, 20990231, and 21074062).

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